Question: Given $ \overrightarrow{OA}\perp\overrightarrow{OC}$, $ m \angle BOC = 7x - 42$, and $ m \angle AOB = 6x - 76$, find $m\angle AOB$. $O$ $A$ $C$ $B$
Explanation: From the diagram, we see that together ${\angle AOB}$ and ${\angle BOC}$ form ${\angle AOC}$ , so $ {m\angle AOB} + {m\angle BOC} = {m\angle AOC}$ Since we are given that $\overrightarrow{OA}\perp\overrightarrow{OC}$ , we know ${m\angle AOC = 90}$ Substitute in the expressions that were given for each measure: $ {6x - 76} + {7x - 42} = {90}$ Combine like terms: $ 13x - 118 = 90$ Add $118$ to both sides: $ 13x = 208$ Divide both sides by $13$ to find $x$ $ x = 16$ Substitute $16$ for $x$ in the expression that was given for $m\angle AOB$ $ m\angle AOB = 6({16}) - 76$ Simplify: $ {m\angle AOB = 96 - 76}$ So ${m\angle AOB = 20}$.